#Fourier series

* Decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials).
* Suppose $f(x)$ can be expressed as some infinite sum: 
	$$ \frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)] $$
* Due to _orthogonality_ of sinusoid functions, we can caluculate the Fourier coefficients of $f(x)$:
    $$ a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0 $$
and
    $$ b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1 $$

[How do we know the complex exponentials "span" the set of all real functions?](http://math.stackexchange.com/questions/128866/how-do-we-know-the-complex-exponentials-span-the-set-of-all-real-functions)

Quote (with slight modification):
> Fix some $N$, and
> consider the subset $S_N$ of $[0,2\pi]$ consisting of the points
> $0, 2\pi/N, 4\pi/N, \ldots, (N-1)2\pi/N$.
> 
> Straightforward linear algebra (_n-dimension vector space_, _systems of
linear equations_, etc) shows that a function on the finite set $S_N$ can be written
> uniquely as a linear combination of 
> the exponentials $e^{i n x}$, for $0 \leq n \leq N-1$.
> 
> Precisely, if $\phi$ is a function on $S_N$, then 
> $\phi(x) = \sum_{n=0}^{N-1} a_n e^{2\pi i n/N},$ (TODO: sth wrong here)
> where
> $$a_n = \dfrac{1}{N}\sum_{n = 0}^{N-1} \phi(x) e^{-2\pi i n/N}.$$

According to [1] Chapter 8, when decomposes a 16-point signal into 9 sine and 9 cosine waves, "the frequency of each sinusoid is fixed; only the amplitude is changed depending on the shape of the waveform being decomposed."

[1]: The Scientist and Engineer's Guide to Digital Signal Processing
#